BMAD

At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two.

In a strange village in cyberspace the inhabitants are "truthers" or liars depending not only on the day of the week but also on whether the day is fair or rainy. Three inhabitants are met. It is known that Amy lies on fair Tuesdays, fair Thursdays, and fair Saturdays, and on rainy Mondays, rainy Wednesdays, and rainy Fridays. At all other times she tells the truth. On the other hand, both Bonnie and Cybil lie on fair Mondays, fair Wednesdays, and fair Fridays, and on rainy Tuesdays, rainy Thursdays, and rainy Saturdays. At all other times they tell the truth. Amy, Bonnie, and Cybil make the following statements. Amy: It is raining and today is Tuesday Bonnie: It is fair or today is Tuesday Cybil: It was not Wednesday yesterday and it will not be Wednesday tomorrow. Questions 1. What day of the week is it? 2. Is it fair or is it raining?

Which of these is larger? the total number of digits in the sequence 1, 2, 3, ..., 101180 or the number of zero digits in the sequence 1, 2, 3, ... , 101181

We have two coins: coin A gives heads with probability 1/3; coin B gives heads with probability 9/10. Pick one of the two, call it C, is selected randomly (using equal probability). Devise a scheme that involves flipping C at most n times, looking at the results, and declaring whether C = A or C = B. We want the declaration to be correct at least 999,999 times out of a million. And we want n to be as small as possible.

You are a competitive swimmer, and your coach wants you to swim a lap of the pool backstroke with a soda can balanced on your forehead. He gives you an empty soda can and you can add some water to it if that will make your task easier. How much water should you place in the can so that the center of gravity of the water-plus-can is as low as possible? Assume the can is a perfect cylinder with a top and bottom made of the same material as the sides. The density of water is 1 gram/centimeter³. Let H be the height of the cylinder and r its radius, and let the mass of the can be C grams.

Fred has 100 lbs of potatoes which consist of 99 percent water. Leaving them outside overnight, Fred finds his potatoes only consist of 98 percent water. What is their new weight? What was the weight of the 1% of water lost? What is the weight of the 98% water remaining?

This trick is an alternative method for multiplying two single digit numbers from 9 to 5. It assumes the user can multiply single digit numbers up to 4. Here is how it works: List 9 to 5 in two lines. Underline the numbers one wants to multiply Everything in the column under the line is worth ten Every digit in the column above the lines are counted. Multiply the counted digits. Add the tens to the products of the digits Voila Why does this work? For example: 9 x 7 9 9 _ 8 8 7 7 _ 6 6 5 5 1d 3d _ 10 10 _ 10 10 10 10 60 + 1 x 3 = 63 ------------------------ 6x5 9 9 8 8 7 7 6 6 _ 5 5 _ 4d 5d _ 10 10 + 4 x 5 = 30

What is the smallest number that is divisible by 2, 3, 4, 5 and 6 with one digit left over, yet is evenly divided by 7? The answer is pretty straightforward but I am interested in seeing if there are various proofs to this problem.

Place the remaining numbers from 4 to 10 in the seven divisions of the above figure so that the outer divisions total 30 and each geometric figure totals 30.

If we use non-integer radii does the inner area approach an irrational number?

sometimes... learning new techniques to solving the familiar is the only award one gets. The result means little to me. I was just interested in seeing how others attacked this problem.

guess they were. Alas, straightforward intuition wins again.

Lol

Suppose you wrote all the letters to the alphabet down in a random sequence. What is the probability that you write down at least 5 letters in proper alphabetical order? For clarity (hopefully): for a letter to be in proper alphabetical order, the letter does not have to be moved or switched to the left past another letter in the given list. When deciding how many letters in a given random list are in order note that Letters need not be in adjacent order A,B,C,D,...) to be in proper alphabetical order. See examples below for more clarity. Examples: A L C E H. .... 4 letters are in proper alphabetical order (by dropping L) ZYXWU ..... 1 letter is in proper letter order, pick one HDAZV ........ 2 letters are in proper letter order, AZ for example

What is the probability that you can pick a natural number that contains the digit 1 in the set of natural numbers?

how would one construct this?

thank you gavinksong. to type with spoilers, you can manually create them by typing "spoiler" in brackets not quotes to start your message, then type "/spoiler" in brackets not quotes when finished.

Place N unit circles (radius=1) together without overlapping. What is the smallest circle which can be circumscribed around the unit circles?

What is the largest circle you can construct in a unit cube?

yes to both of your questions.

What is the median area found between three tangential circles when the radius of the three circles are between 1inch and 5 inch inclusive?

corners count means that diagonal adjacency is permitted. And yes, only letter characters.

On a standard qwerty keyboard, how many "2-letter adjacency words" can be formed? Adjacent implies that the word must be formed by two distinct letters that are next to each other on a keyboard (corners count) Any two letters, for the sake of argument, form a two-letter word.